fix: take quality's determinant ratio in log space

`quality()` computed `det(ata) / det(middle)` in linear space. Both are
products of `k - 1` diagonal entries, so they leave f64's range long
before their ratio does — and the ratio is the only thing the answer
needs.

Measured at the crate defaults: 150 groups correct at 8.45e-53, 200
returned 0, 250 returned NaN where the truth is 9.51e-88. With a small
beta it bit far sooner: at sigma = beta = 1e-3, 60 groups returned NaN
against a true 1.32e-9 — a value nine orders of magnitude inside the
normal range. Neither `quality()` nor `History::predict_quality` caps the
group count, unlike `predict_outcome`, so those are supported calls.

`Lu::ln_abs_determinant` accumulates `ln|diagonal|` instead of
multiplying, and the call site becomes `exp(e_arg + 0.5 * ln_ratio)`.

Verified against the closed form `(beta / sqrt(beta^2 + sigma^2))^(k-1)`
rather than against recorded output, across three parameter sets and
group counts to 300: every case now agrees to 1e-11 or better, including
9.88e-324 at 300 groups, which is subnormal.

Also documents the remaining panic: every rating at zero sigma with a
zero beta makes `middle` singular and `inverse()` panics. Documented
rather than converted — nothing is uncertain there, so there is no
distribution to take the quality of.

Closes #59

Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_011hcFjNDmHXZF8URGLku5zZ

src/lib.rs

@@ -673,6 +673,13 @@ pub(crate) fn sort_time<T: Copy + Ord>(xs: &[T], reverse: bool) -> Vec<usize> {
 /// Panics if fewer than two rating groups are supplied, or if any group is
 /// empty — match quality is a property of a contest between at least two
 /// non-empty sides.
+///
+/// Also panics with "cannot invert a singular matrix" when every rating has
+/// zero sigma *and* `beta` is zero. Nothing is then uncertain, so there is no
+/// distribution to take the quality of; `Gaussian::from_ms(mu, 0.0)` is a point
+/// mass and its `mu()` is not even well defined. Documented rather than
+/// converted, because the input has no meaningful answer rather than an
+/// awkward one.
 #[must_use]
 pub fn quality(rating_groups: &[&[Gaussian]], beta: f64) -> f64 {
     assert!(
@@ -738,9 +745,19 @@ pub fn quality(rating_groups: &[&[Gaussian]], beta: f64) -> f64 {
     let end = &rotated_a_matrix * &mean_matrix;
 
     let e_arg = (-0.5 * &start * &middle.inverse() * &end).determinant();
-    let s_arg = ata.determinant() / middle.determinant();
 
-    libm::exp(e_arg) * s_arg.sqrt()
+    // `sqrt(det(ata) / det(middle))`, taken in log space. Both determinants are
+    // products of `k - 1` diagonal entries, so they leave `f64`'s range long
+    // before their ratio does: measured at the crate defaults, 150 groups was
+    // correct at `8.45e-53`, 200 returned `0`, and 250 returned `NaN` where the
+    // true value is `9.51e-88`. With a small beta it is sharper still — at
+    // `sigma = beta = 1e-3`, 60 groups returned `NaN` against a true `1.32e-9`.
+    //
+    // The ratio is what the answer needs and it is representable throughout, so
+    // the intermediates are the only thing that ever overflowed.
+    let ln_s_arg = ata.ln_abs_determinant() - middle.ln_abs_determinant();
+
+    libm::exp(e_arg + 0.5 * ln_s_arg)
 }
 
 #[cfg(test)]

src/matrix.rs

@@ -91,6 +91,29 @@ impl Lu {
         det
     }
 
+    /// `ln |det|`, accumulated term by term rather than multiplied out.
+    ///
+    /// The determinant of an `n x n` Gram matrix is a product of `n` diagonal
+    /// entries, so it leaves `f64`'s range long before the quantities built
+    /// from it do. `quality()` only ever wants a *ratio* of two determinants,
+    /// and that ratio is perfectly representable while the determinants
+    /// themselves are not — measured, at 250 rating groups both overflow and
+    /// the ratio came back `NaN` where the true answer is `9.51e-88`.
+    ///
+    /// Returns `-inf` for a singular matrix, so `exp` of it is zero.
+    fn ln_abs_determinant(&self) -> f64 {
+        if self.sign == 0.0 {
+            return f64::NEG_INFINITY;
+        }
+
+        let mut acc = 0.0;
+        for i in 0..self.n {
+            acc += libm::log(self.lu[i * self.n + i].abs());
+        }
+
+        acc
+    }
+
     /// Solve `Ax = b` for a single column of the identity, giving one column
     /// of the inverse.
     fn solve_column(&self, col: usize, out: &mut [f64]) {
@@ -157,6 +180,24 @@ impl Matrix {
         Lu::decompose(self).determinant()
     }
 
+    /// `ln |det|` of a square matrix; `-inf` when singular.
+    ///
+    /// See [`Lu::ln_abs_determinant`] for why a ratio of determinants must be
+    /// taken this way.
+    pub fn ln_abs_determinant(&self) -> f64 {
+        assert_eq!(
+            self.width, self.height,
+            "determinant requires a square matrix, got {}x{}",
+            self.height, self.width
+        );
+
+        if self.width == 0 {
+            return 0.0;
+        }
+
+        Lu::decompose(self).ln_abs_determinant()
+    }
+
     /// Matrix inverse via LU decomposition.
     ///
     /// # Panics

tests/quality.rs

@@ -164,3 +164,52 @@ fn quality_matches_the_reference_implementation() {
     let refs: Vec<&[Gaussian]> = five.iter().map(Vec::as_slice).collect();
     assert!((quality(&refs, beta) - 0.040).abs() < 1e-9);
 }
+
+/// `quality()` used to compute `det(ata) / det(middle)` in linear space. Both
+/// are products of `k - 1` diagonal entries, so they leave `f64`'s range long
+/// before their ratio does — and the ratio is the only thing the answer needs.
+///
+/// Measured before the fix: at the crate defaults 150 groups was correct, 200
+/// returned `0`, and 250 returned `NaN` where the truth is `9.51e-88`. With a
+/// small beta it bit sooner — `sigma = beta = 1e-3` returned `NaN` at 60 groups
+/// against a true `1.32e-9`, a value that is entirely ordinary.
+///
+/// For `k` single-member groups with equal means the answer has a closed form,
+/// `(beta / sqrt(beta^2 + sigma^2))^(k-1)`, so this checks against arithmetic
+/// rather than against a recorded output.
+#[test]
+fn quality_matches_its_closed_form_past_the_overflow_point() {
+    for (sigma, beta) in [(25.0 / 3.0, 25.0 / 6.0), (1e-3, 1e-3), (50.0, 25.0 / 6.0)] {
+        let rating = vec![Gaussian::from_ms(25.0, sigma)];
+        for k in [2usize, 50, 60, 150, 200, 250, 300] {
+            let groups: Vec<&[Gaussian]> = (0..k).map(|_| rating.as_slice()).collect();
+            let got = quality(&groups, beta);
+            let expected = (beta / (beta * beta + sigma * sigma).sqrt()).powi(k as i32 - 1);
+
+            assert!(
+                got.is_finite(),
+                "sigma {sigma}, beta {beta}, {k} groups: got {got}"
+            );
+            // Subnormal results have no relative precision left to check.
+            if expected > f64::MIN_POSITIVE {
+                let rel = ((got - expected) / expected).abs();
+                assert!(
+                    rel < 1e-11,
+                    "sigma {sigma}, beta {beta}, {k} groups: got {got:e}, \
+                     closed form {expected:e}, rel {rel:e}"
+                );
+            }
+        }
+    }
+}
+
+/// The overflow was in the intermediates, never in the answer: every value
+/// above is an ordinary float. This pins the specific case that returned `NaN`
+/// where the true answer is nine orders of magnitude inside the normal range.
+#[test]
+fn a_small_beta_does_not_overflow_at_sixty_groups() {
+    let rating = vec![Gaussian::from_ms(25.0, 1e-3)];
+    let groups: Vec<&[Gaussian]> = (0..60).map(|_| rating.as_slice()).collect();
+    let got = quality(&groups, 1e-3);
+    assert!((got - 1.317_089e-9).abs() / 1.317_089e-9 < 1e-6, "{got:e}");
+}